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Discontinuous Galerkin finite element approximation of quasilinear elliptic boundary value problems I: The scalar case. (English) Zbl 1084.65116

The quasilinear elliptic equation \[ -\nabla\{\mu(x,| \nabla u| )\nabla u =f \] is considered under assumptions on the function \(\mu\) such that existence and uniqueness are obtained by the theory of monotone operators. The corresponding results are established by discontinuous Galerkin finite elements with a one-parameter family of stabilization. If \(u\in C^1(\Omega)\cap H^k(\Omega)\), \(k\geq2\), then with discontinuous piecewise polynomials of degree \(p\geq1\), the error between \(u\) and \(u_{DG}\) measured in the broken \(H^1(\Omega)\)-norm, is \({\mathcal O}(h^{s-1}/p^{k-3/2})\), where \(1\leq s\leq \min\{p+1,k\}\).

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations
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